Question: The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives $23.2$ years; the standard deviation is $4.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a gorilla living between $8.5$ and $13.4$ years.
$23.2$ $18.3$ $28.1$ $13.4$ $33$ $8.5$ $37.9$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $23.2$ years. We know the standard deviation is $4.9$ years, so one standard deviation below the mean is $18.3$ years and one standard deviation above the mean is $28.1$ years. Two standard deviations below the mean is $13.4$ years and two standard deviations above the mean is $33$ years. Three standard deviations below the mean is $8.5$ years and three standard deviations above the mean is $37.9$ years. We are interested in the probability of a gorilla living between $8.5$ and $13.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the gorillas will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the gorillas will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of gorillas between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular gorilla living between $8.5$ and $13.4$ years is $\color{orange}{2.35\%}$.